drill: mon, 10/27 1. write a conditional statement for the statement “all ravens fans are from...
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Drill: Mon, 10/27
1. Write a conditional statement for the statement “All Ravens fans are from Maryland”
2. Write the converse of your statement.
3. Write the contrapositive of the conditional.
OBJ: SWBAT write and analyze biconditional statements.
Get out your Practice A homework!
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UNIT C (2.4)BICONDITIONAL STATEMENTS
Honors Geometry
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EXAMPLES (WRITE THE CONVERSE, INVERSE AND CONTRAPOSITIVE FOR EACH)
If it snows six feet, then schools will be closed. Converse: If schools will be closed, then it snowed six
feet.Inverse: If it does not snow six feet, then schools will
not be closed.Contrapositive: If schools will not be closed, then it did
not snow six feet.
If it is December 25th, then it is Christmas Day. Converse: If it is Christmas Day, then it is December
25th.Inverse: If it is not December 25th, then it is not
Christmas Day.Contrapositive: If it not Christmas Day, then it is not
December 25th.
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When you combine a conditional statement and its converse, you create a biconditional statement.
A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.”
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Holt McDougal Geometry
2-4 Biconditional Statements and Definitions
p q means p q and q p
The biconditional “p if and only if q” can also be written as “p iff q” or p q.
Writing Math
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Write the conditional statement and converse within the biconditional.
Example 1A: Identifying the Conditionals within a Biconditional Statement
An angle is obtuse if and only if its measure is greater than 90° and less than 180°.
Conditional: If an is obtuse, then its measure is greater than 90° and less than 180°.
Converse: If an angle's measure is greater than 90° and less than 180°, then it is obtuse.
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Write the conditional statement and converse within the biconditional.
Example 1B: Identifying the Conditionals within a Biconditional Statement
A solution is neutral its pH is 7.
Conditional: If a solution is neutral, then its pH is 7.
Converse: If a solution’s pH is 7, then it is neutral.
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Check It Out! Example 1a
An angle is acute iff its measure is greater than 0° and less than 90°.
Write the conditional statement and converse within the biconditional.
Conditional: If an angle is acute, then its measure is greater than 0° and less than 90°.
Converse: If an angle’s measure is greater than 0° and less than 90°, then the angle is acute.
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Check It Out! Example 1b
Cho is a member if and only if he has paid the $5 dues.
Write the conditional statement and converse within the biconditional.
Conditional: If Cho is a member, then he has paid the $5 dues.
Converse: If Cho has paid the $5 dues, then he is a member.
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For each conditional, write the converse and a biconditional statement.
Example 2: Identifying the Conditionals within a Biconditional Statement
A. If 5x – 8 = 37, then x = 9.
Converse: If x = 9, then 5x – 8 = 37.
B. If two angles have the same measure, then they are congruent.
Converse: If two angles are congruent, then they have the same measure.
Biconditional: 5x – 8 = 37 if and only if x = 9.
Biconditional: Two angles have the same measure if and only if they are congruent.
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Check It Out! Example 2a
If the date is July 4th, then it is Independence Day.
For the conditional, write the converse and a biconditional statement.
Converse: If it is Independence Day, then the date is July 4th.
Biconditional: It is July 4th if and only if it is Independence Day.
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Check It Out! Example 2b
For the conditional, write the converse and a biconditional statement.
If points lie on the same line, then they are collinear.
Converse: If points are collinear, then they lie on the same line.
Biconditional: Points lie on the same line if and only if they are collinear.
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For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false.
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Determine if the biconditional is true. If false, give a counterexample.
Example 3A: Analyzing the Truth Value of a Biconditional Statement
A rectangle has side lengths of 12 cm and 25 cm if and only if its area is 300 cm2.
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Example 3A: Analyzing the Truth Value of a Biconditional Statement
Conditional: If a rectangle has side lengths of 12 cm and 25 cm, then its area is 300 cm2.
Converse: If a rectangle’s area is 300 cm2, then it has side lengths of 12 cm and 25 cm.
The conditional is true.
The converse is false.
If a rectangle’s area is 300 cm2, it could have side lengths of 10 cm and 30 cm. Because the converse is false, the biconditional is false.
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Determine if the biconditional is true. If false, give a counterexample.
Example 3B: Analyzing the Truth Value of a Biconditional Statement
A natural number n is odd n2 is odd.
Conditional: If a natural number n is odd, then n2 is odd.
The conditional is true.
Converse: If the square n2 of a natural number is odd, then n is odd.
The converse is true.
Since the conditional and its converse are true, the biconditional is true.
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Check It Out! Example 3a
An angle is a right angle iff its measure is 90°.
Determine if the biconditional is true. If false, give a counterexample.
Conditional: If an angle is a right angle, then its measure is 90°.
The conditional is true.
Converse: If the measure of an angle is 90°, then it is a right angle.
The converse is true.
Since the conditional and its converse are true, the biconditional is true.
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Check It Out! Example 3b
y = –5 y2 = 25
Determine if the biconditional is true. If false, give a counterexample.
Conditional: If y = –5, then y2 = 25. The conditional is true.
Converse: If y2 = 25, then y = –5.
The converse is false.
The converse is false when y = 5. Thus, the biconditional is false.
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In geometry, biconditional statements are used to write definitions.
A definition is a statement that describes a mathematical object and can be written as a true biconditional.
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In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments.
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A triangle is defined as a three-sided polygon, and a quadrilateral is a four-sided polygon.
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A GOOD DEFINITION?
A square is a parallelogram.
A square is a parallelogram with four right angles.
Definitions must be reversible in order to be correct.
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Think of definitions as being reversible. Postulates, however are not necessarily true when reversed.
Helpful Hint
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Write each definition as a biconditional.
Example 4: Writing Definitions as Biconditional Statements
A. A pentagon is a five-sided polygon.
B. A right angle measures 90°.A figure is a pentagon if and only if it is a 5-sided polygon.
An angle is a right angle if and only if it measures 90°.
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Check It Out! Example 4
4a. A quadrilateral is a four-sided polygon.
4b. The measure of a straight angle is 180°.
Write each definition as a biconditional.
A figure is a quadrilateral if and only if it is a 4-sided polygon.
An is a straight if and only if its measure is 180°.
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PRACTICE WITH BICONDITIONALS
pp. 6 – 7, 20 – 21, 28, and 50
Work in pairs to rewrite the definitions as biconditional statements.
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PRACTICE A
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Lesson Quiz
1. For the conditional “If an angle is right, then its measure is 90°,” write the converse and a biconditional statement.
2. Determine if the biconditional “Two angles are complementary if and only if they are both acute” is true. If false, give a counterexample.
False; possible answer: 30° and 40°
Converse: If an measures 90°, then the is right. Biconditional: An is right iff its measure is 90°.
3. Write the definition “An acute triangle is a triangle with three acute angles” as a biconditional.
A triangle is acute iff it has 3 acute s.